Quantum-optical spectroscopy of a two-level system using an electrically driven micropillar laser as resonant excitation source

Two-level emitters constitute main building blocks of photonic quantum systems and are model systems for the exploration of quantum optics in the solid state. Most interesting is the strict-resonant excitation of such emitters to generate close to ideal quantum light and to control their occupation coherently. Up till now related experiments have been performed exclusively using bulky lasers which hinders the application of resonantly driven two-level emitters in photonic quantum systems. Here we perform quantum-optical spectroscopy of a two-level system using a compact high-$\beta$ microlaser as excitation source. The two-level system is based on a semiconductor quantum dot (QD), which is excited resonantly by a fiber-coupled electrically driven micropillar laser. In this way we dress the excitonic state of the QD under continuous wave excitation and trigger the emission of single-photons with strong multi-photon suppression ($g^{(2)}(0)=0.02$) and high photon indistinguishably ($V=57\pm9\%$) via pulsed resonant excitation at 156 MHz.

W a v e l e n g t h ( n m ) Figure S1. Overview µEL spectrum of the micropillar laser under CW excitation. The spectrum is recorded at the output of the single-mode fiber and shows pronounced single-mode emission of the fundamental laser mode. We would like to point out that no significant contribution from GaAs bandgap or wetting layer emission is detected. F r e q u e n c y ( G H z ) 6 . 7 5 m W Figure S2. Resonance fluorescence spectra of the QD studied in the main text for tunable-laser excitation with five different excitation powers from 17 µW to 6.75 mW.
The sidebands of the characteristic excitation power dependent splitting of the Mollow-triplet is clearly observed. Symbols present measured spectra and solid curves correspond to calculated spectra obtained by fitting the experimental data according to Eq. S3. A constant offset is introduced for the sake of clarity.  Figure S3. 2D contour plot of excitation power dependent resonance fluorescence spectra. The frequency distance of the the Mollow triplet sidebands to the carrier frequency divided by the square root of the excitation power is 1.33 THz/ √ W.  Figure S4. µPL intensity of micropillar laser in pulsed operation as a function of V bias and V peak = 8 V for an electrical pulse-width of 520 ps and a pulse separation of 6.4 ns. The green line indicates the chosen DC bias (V bias = 4.7 V) chosen for the auto-correlation and HOM-measurements presented in Fig. 5 of the main text.

SIII. Emission properties of the micropillar laser under pulsed electrical excitation
As mentioned in the main part of the manuscript, single photons were triggered by exciting the QD with short resonant micropillar laser pulses. To generate these laser pulses, the supply voltage of the micropillar laser triggered electrically. In the applied excitation scheme the laser was biased with V bias = 4.7 V on top of which pulses with a peak voltage of V peak = 8 V and width of 520 ps FWHM were added. The pulse period is 6.4 ns corresponding to a repetition frequency of 156 MHz. Fig. S4 presents the output intensity of the microlaser as a function of the DC bias for constant pulse hight of 8 V. Fig. S5 displays the time resolved emission intensity of the microlaser in logarithmic scale and Fig. S6 shows the second-order auto-correlation function g (2) (τ ) of the micropillar, both V peak = 8 V. As expect for coherent laser emission g (2) (0) = 1.016(12).   Figure S6. Second-order photon auto-correlation function g (2) (τ ) of the pulsed driven QD micropillar laser. From the data we extract g (2) (0) = 1.016(12) indicating coherent laser emission.

SIV. Theoretical description of resonance fluorescence
we treat the QD as a two-level system with the ground state |g and the exited state |e , driven by a laser light-field oscillating with the angular frequency ω L with the time-dependent envelope of arbitrary phase Ω(t). For resonant driving the system can be described via the rotating wave approximation and in the rotating frame of ω L by the Hamiltonian with the annihilation and creation operators σ − = |g e| and σ + = |e g|. The numerical simulation relies on the quantum-optical master equation in the Lindblad form 1 With the decay rate Γ given by the inverse of the excitonic lifetime T 1 and the pure dephasing rate γ given by γ = 1 T2 − Γ 2 .

SV. Theoretical description of the Hong-Ou-Mandel experiment under pulsed resonant excitation
In order to describe the experimental data presented in Fig. 5 of the main manuscript we use the master equation introduced in Sec. SIV. The envelope Ω(t) of the driving laser light field is given by a pulse train of Gaussian pulses with pulse separation δ = 6.4 ns. Each pulse is of the FWHM w and covers a pulse area A: Following Fischer et al., 2 we apply the quantum regression theorem to numerically calculate the correlation functions The second-order autocorrelation function g (2) (τ ) is proportional to G (2) (τ ) and since in experimental practice often no strict difference is made between g (2) and G (2) , the calculation of G (2) (τ ) is sufficient.
For measuring the quantum-mechanical two-photon interference of two successive photons originating from the same source, a Mach-Zender interferometer is placed before the actual Hong-Ou-Mandel beam splitter. 3 The crosscorrelogram of this experiment in cross-and co-polarized configuration is calculated similar to 4 but additionally for arbitrary beam-splitter transmission / reflection ratios, yielding where Θ 1/2 and R 1/2 are the transmitivity and reflectivity of the first / second beamsplitter and ∆ is the delay corresponding to the arm-length difference of the Mach-Zehnder interferometer.

SVI. Theoretical description of resonance fluorescence under CW excitation
The quantum optical master equation (see Sec. SIV) leads to the Bloch equation, which -in case of CW excitation Ω(t) = Ω -can be cast conviently in the following form: with the abbreviations: With the inversion operator σ z . Now, it is straight forward to solve these differential equations via Laplace transformation or matrix diagonalization.

Power spectrum and Mollow triplet
For deriving the theoretical formula of the power spectrum, predicting the Mollow-triplet presented in Fig. 4, the first-order correlation function needs to be computed: so the time-dynamics of the expectation value of transition operator is necessary, which reads using following abbreviations For all possible initial conditions, the solution allow the calculation of all superposition of the exciton dynamics and therefore also the two-time correlation function in the steady-state limit: lim Therefore, the steady state values of the system are needed: χ = σ + ∞ = −iΩA 3 /Γ P and σ z ∞ = A 3 . Via the quantum regression theorem, the dynamics of the polarization dynamics is given with different initial values, such as . This leads to the following correlation function dynamics: Now, given this dynamics, the spectrum can easily be calculated, and the formula in the main text is derived with the abbreviation: Changing back from the rotating frame to the laboratory frame and applying the single-sided Fourier transform Eq. S2, the result for the power spectrum presented in Fig. 4 is given by Two-photon autocorrelation function g (2) (τ ) The theoretical description of the photon-autocorrelation function was derived straight-forward from Eq. S1 by diagonalization and subsequent application of the quantum regression theorem. The result is in agreement with the ansatz published in Ref. 5 and is given by  Table I. Equal-time two-photon autocorrelation function without (g (2) (0)) and with (g (2) (0)) blinking compensation and blinking-factor B. Due to blinking of the QD, the g (2) (τ ) is increased by a factor B for small time delays τ . Hence, the blinking-compensated autocorrelation is calculated by normalizing the original function to 1 in the vicinity of zero time delay: where an excitation power dependent factor B > 1 accounts for blinking of the QD. Only in the short-delay limit, B can be approximated to be constant, thus the limitation of the domain to τ ∈ [−4 ns, 4 ns]. For longer time delays, B is expected to decrease and eventually converge to B = 1 in the infinite-delay limit. Table I lists the equaltime two-photon autocorrelation functions without (g (2) (0)) and with (g (2) (0)) blinking compensation as well as the blinking-factor B.
SVII. Hong-Ou-Mandel visibility analysis, compensating non-vanishing g (2) (τ = 0) In a Hong-Ou-Mandel (HOM) style two-photon quantum-interference experiment, multi-photon pulses entering the beamsplitter lead to an increase of equal-time coincidences, even if the photons are indistinguishable. Thus an g (2) (τ = 0) > 0 masks the quantum-interference and, if not taken into account properly, leads to an underestimation of the photon indistinguishability in terms of the visibility V (c.f. Eq. 1 in the main manuscript). In order to extract the pure HOM-visibility V pure from the experimental data presented in Fig. 5b we follow the procedure described in Ref.. 6 Considering our experimental HOM-configuration presented in Fig. 1 of the main text with a pulse-separation of δ (pulse rate: 1 δ ), we obtain peak-areas A i (while neglecting a proportionality constant w.r.t. the total detected counts) for the coincidence peaks: We further define: and: where we have denoted the cross-polarized reference measurements as ⊥ and the co-polarized measurements as .
Eventually we obtain for the pure HOM-visibility V pure : A good matching of the Mach-Zehnder delay to the pulse repetition rate is essential to achieve high visibilities. However our experimental setup was well inside a |∆ − δ| < 20 ps window.
Based on these considerations we able to determine the pure HOM-visibility corrected for non-ideal multi-photon suppression (see Fig. 5a). Analysing the experimental data presented in Fig. 5b by taking into account Gaussian propagation of uncertainty with the main contribution originating from the shot noise of the peak areas we obtain the pure HOM-Visibility V pure = 57(9)%.

SVIII. Parameters affecting the Hong-Ou-Mandel visibility
In the following we discuss effects which influence the HOM-visibility and the related photon-indistinguishably in our experimental setting. As mentioned in the main text, the spontaneous lifetime T 1 = 510 ps, the QD coherence time T 2 = 500 ps, the laser pulse-width w = 200 ps and the laser pulse-area Ω(t)dt = 0.9 π were determined from experimental data. Furthermore, the delay ∆ in the Mach-Zehnder interferometer of the HOM-configuration was optimized to precisely match the pulse distance δ, i.e. |∆ − δ| < 20 ps.
In order to explain the non-ideal HOM visibility and how it can be improved, we calculate the cross-correlogram numerically for different values of the parameters w, ∆ and T 2 as described in Sec. SV and determine V and V pure by Eqs. 1 and S5. The results are presented in Figs. S7, S8, S9and S10(only for V ), respectively.
The increased pulse-width in our experiment (compared to standard experiments applying excitation pulses with ps pulse-width) has moderate direct influence on the HOM-visibility and leads to a reduction from 47% to 42% for the given experimental parameters. As can be seen in Fig. S8 a delay mismatch would be more detrimental, but this parameter is well under control in experiment with |∆ − δ| < 20 ps. Most problematic is the coherence time which limits the HOM-visibility to values below 50% in the present case (see Fig. S9). Since Fourier-limited linewidths and close to ideal HOM-visibility has been performed for QDs based on the same wafer material we assume that the increase pulse-width leads to enhanced dephasing of the resonantly driven QD exciton, thereby reducing the HOMvisibility. Thus, further optimization need to focus on improved pulse-schemes to improve the T 2 /(2T 1 ) towards the Fourier limit of 1 in order to enhance the HOM-visibility in the applied excitation scheme.  Figure S10. Simulation of both QD coherence time T2 dependence and pulse width w dependence of HOM visibility with ∆ − δ = 0. The black cross marks the experimental parameters. Obviously a reduction of the pulse width is only significantly beneficial in the almost fourierlimited regime where T2 is almost equal to 2T1.